Natural Deduction with Propositional Logic

Introducing Natural Natural Deduction with Propositional Logic Deduction Ling 130: Formal Semantics Some basic rules without assumptions

Rules with assumptions

Spring 2018 Outline

Natural Deduction with Propositional Logic

Introducing 1 Introducing Natural Deduction Natural Deduction

Some basic rules without assumptions 2 Some basic rules without assumptions Rules with assumptions

3 Rules with assumptions What is ND and what’s so natural about it?

Natural Deduction with Natural Deduction Propositional Logic A system of logical proofs in which assumptions are freely introduced but discharged under some conditions.

Introducing Natural Deduction Introduced independently and simultaneously (1934) by Some basic Gerhard Gentzen and Stanislaw Ja´skowski rules without assumptions

Rules with assumptions

The book & slides/handouts/HW represent two styles of one ND system: there are several. Introduced originally to capture the style of reasoning used by mathematicians in their proofs. Ancient antecedents

Natural Deduction with Propositional Logic Aristotle’s syllogistics can be interpreted in terms of inference rules and proofs from assumptions.

Introducing Natural Deduction

Some basic rules without assumptions Rules with Stoic logic includes a practical application of a ND assumptions theorem. ND rules and proofs

Natural Deduction with Propositional There are at least two rules for each connective: Logic an introduction rule an elimination rule Introducing Natural The rules reflect the meanings (e.g. as represented by Deduction

Some basic truth-tables) of the connectives. rules without assumptions

Rules with Parts of each ND proof assumptions You should have four parts to each line of your ND proof: line number, the formula, justification for writing down that formula, the goal for that part of the proof. # formula justification rule Goal 1 A we proved this... Goal: something else Conjunction introduction

Natural Deduction with &-introduction, or &I Propositional Logic To introduce a conjunction formula, you have to have already introduced both conjuncts. Introducing Natural Deduction Some basic 1. A we proved this... Goal:A&B rules without assumptions . Rules with 2. . (other junk) assumptions 3. B and this... 4. A&B &I (1, 3)

because that’s the only way the conjunction would be true, when both conjuncts are true. Conjunction elimination

Natural Deduction with Propositional &-elimination, or &E Logic If you managed to write down a conjunction, you can go on to

Introducing write down one or both conjuncts by themselves. Natural Deduction

Some basic rules without assumptions 1. A&B we proved this somehow Goal: A, B Rules with assumptions 2. B &E(1) 3. A &E(1)

because if conjunction is true, you’re in the first line of the truth-table, so each conjunct is true. Disjunction introduction (Monkey’s uncle!)

Natural Deduction with Propositional Logic Truth of one disjunct is sufficient to make the whole disjunction true.

Introducing Natural Deduction ∨-introduction, or ∨I

Some basic rules without To introduce a disjunction formula, you have to have one of assumptions the disjuncts. Rules with assumptions B can be any formula at all!

1. A we proved this... Goal:A ∨ B 2. A ∨ B ∨I (1) Disjunction introduction (Monkey’s uncle!)

Natural Deduction with Propositional Logic Truth of one disjunct is sufficient to make the whole disjunction true.

Introducing Natural Deduction ∨-introduction, or ∨I

Some basic rules without To introduce a disjunction formula, you have to have one of assumptions the disjuncts. Rules with assumptions Or, if you want, A can be any formula at all!

1. B we proved this... Goal:A ∨ B 2. A ∨ B ∨I (1) Conditional elimination (Modus Ponens)

Natural Deduction with →-elimination, E or Modus Ponens, MP Propositional Logic If you have a conditional formula, and, separately, the formula matching the antecedent of the conditional, you can write Introducing down the formula matching the consequent of the conditional. Natural Deduction

Some basic 1. A → B we proved this somehow... Goal:B rules without assumptions 2. . (other junk) Rules with . assumptions 3. A and this... 4. B → E(1, 3)

conditional=1, so not the 2nd line of that truth-table. antecedent=1, so not lines 3, 4 of that truth-table. So you’re in the 1st line, so consequent=1. Negation elimination, ¬E

Natural Deduction ¬-elimination, or ¬E, or Contradiction rule with Propositional Logic You’re not actually eliminating the negation: when you have both a formula and its negation, it’s a contradiction.

Introducing Natural Deduction 1. ¬A we proved this... Goal:⊥ Some basic rules without assumptions . 2. . (other junk) Rules with assumptions 3. A and this... 4. ⊥ ¬E(1, 3) / Contradiction(1, 3)

Negation truth-table: no line where A = 1 and ¬A = 1 Why would you ever want to derive a contradiction? We’ll see shortly! Negation elimination, ¬E

Natural Deduction with ¬-elimination, or ¬E, or Contradiction rule Propositional Logic You’re not actually eliminating the negation: when you have both a formula and its negation, it’s a contradiction. Introducing Natural Deduction I find it clearer to do this, but you can choose: Some basic rules without assumptions 1. ¬A we proved this... Goal:⊥ Rules with assumptions . 2. . (other junk) 3. A and this... 4. A ∧ ¬A ∧I (1, 3) 5. ⊥ ¬E(4) / Contradiction(4) Double negation rules

Natural Deduction with ¬¬-elimination, or ¬¬E Propositional Logic Double negative makes a positive.

Introducing Natural Deduction 1. ¬¬A we got this... Goal: A

Some basic rules without 2. A ¬¬E assumptions

Rules with assumptions ¬¬-introduction, or ¬¬I Positives makes a double negative.

1. A we got this... Goal: ¬¬A 2. ¬¬A ¬¬I Rules and proofs with assumptions

Natural Deduction with Propositional Writing proofs with assumptions Logic You can assume anything you want, as long as you know how

Introducing to get rid of that assumption later (discharge it). Natural Deduction Starting with the assumption, and until the assumption is

Some basic discharged, lines/formulas are enclosed in a box. rules without assumptions Rules with During the proof, you can use your assumption as well as assumptions anything already established. But you can’t use that assumption or anything that depends on that assumption later. This means that once the box is closed, all the stuff inside the box becomes unusable. Rules and proofs with assumptions

Natural Deduction with Propositional Logic 1. stuff this Goal: something

Introducing Natural 2. A as. Box goal: blah Deduction

Some basic 3. proof hard struggle rules without assumptions 4. blah we made it! Rules with assumptions 5. B assumption discharged

Box goal often differs from goals outside the box. While writing lines (2-4), we can use stuff from line 1. From line 5 on, we cannot use formulas from lines (2-4) Rules and proofs with assumptions

Natural Deduction with Propositional Logic Typesetting note: Introducing Boxes have to include at least the actual formulas; Natural Deduction optionally also the explanations, goals, and/or line Some basic numbers. rules without assumptions 1. stuff this Goal: something Rules with assumptions 2. A as. Box goal: blah 3. proof hard struggle 4. blah we made it! 5. B assumption discharged Conditional introduction, → I

Natural Deduction →-introduction (→ I ), or Conditional/Hypothetical proof with Propositional Logic To prove a conditional A → B, show that A entails B: suppose that A and show that B follows.

Introducing Natural Deduction Some basic Goal :A → B rules without assumptions

Rules with assumptions 1. A as. Box goal: B 2. proof hard struggle 3. B we made it!

4. A → B → I (1, 3) Negation introduction, ¬I

Natural Deduction Reductio ad absurdum (proof by contradiction), RAA, or with Propositional ¬-introduction (¬I ) Logic To prove that ¬A is true, show that A entails a contradiction,

Introducing so A must be false. Natural Deduction

Some basic rules without assumptions 1. ... Goal: ¬A Rules with assumptions 2. A as. Box goal: ⊥ 3. proof struggle subgoal: q, ¬q 4. ⊥ yay, A means trouble!

5. ¬A RAA(1, 3) Negation introduction, ¬I

Natural Deduction with Propositional Logic To get the contradiction, we’ll need ¬E rule: find any formula q and its negation ¬q, together they entail ⊥

Introducing Some creativity is required to come up with q that works. Natural Deduction 1. ... Goal: ¬A Some basic rules without assumptions 2. A as. Box goal: ⊥ Rules with assumptions 3. proof struggle subgoal: q, ¬q 4. ⊥ yay, A means trouble!

5. ¬A RAA(1, 3) Proof by cases

Natural Deduction with Propositional Proof by cases, PBC, or ∨-elimination (∨E) Logic To prove something from A ∨ B, you need to prove it by

Introducing assuming A, AND prove it by assuming B. Natural Deduction Some basic I hate the name ∨E, since you are not eliminating rules without assumptions anything. You are just proving that the conclusion follows Rules with in either case. assumptions You know A ∨ B is true, but you don’t know why it’s true: could be because A is true, could be because of B, or both. Since A ∨ B means at least one of the disjuncts is true, showing that the conclusion follows in either case proves that it follows from A ∨ B Proof by cases

Natural Deduction with Propositional Logic We don’t know that A is true, or that B is true (just that at least one of them is) Introducing So, A and B are assumptions, not usable later, so are put Natural Deduction in (side-by-side) boxes. Some basic rules without 1. A ∨ B we got this somehow Goal: C assumptions

Rules with assumptions 2. A as. Box goal: C 5. B as. Box goal: C . . 3. . 1st proof 6. . 2nd proof 4. C got it once! 7. C got it again!

8. C PBC(1, 2 − 4, 5 − 7)